Mon. Not. R. Astron. Soc. 000, 000–000 (0000)
Printed 10 July 2012
(MN LATEX style file v2.2)
Theoretical dark matter halo density profile
Eduard Salvador-Solé⋆, Jordi Viñas, Alberto Manrique and Sinue Serra
arXiv:1104.2334v4 [astro-ph.CO] 9 Jul 2012
Institut de Ciències del Cosmos, Universitat de Barcelona (UB–IEEC), Martı́ i Franquès 1, E-08028 Barcelona, Spain
10 July 2012
ABSTRACT
We derive the density profile for collisionless dissipationless dark matter haloes in hierarchical cosmologies making use of the Secondary Infall (SI) model. The novelties
are: i) we deal with triaxial virialised objects; ii) their seeds in the linear regime are
peaks endowed with unconvolved spherically averaged density profiles according to the
peak formalism; iii) the initial peculiar velocities are taken into account; and iv) accreting haloes are assumed to develop from the inside out, keeping the instantaneous
inner system unaltered. The validity of this latter assumption is accurately checked by
comparing analytical predictions on such a growth with the results of numerical simulation. We show that the spherically averaged density profile of virialised objects can
be inferred with no need to specify their shape. The typical spherically averaged halo
density profile is inferred, down to arbitrarily small radii, from the power-spectrum
of density perturbations. The predicted profile in the ΛCDM cosmology is approximately described by an Einasto profile, meaning that it does not have a cusp but
rather a core, where the inner slope slowly converges to zero. Down to one hundredth
the total radius, the profile has the right NFW and Einasto forms, being close to the
latter down to a radius of about four orders of magnitude less. The inner consistency
of the model implies that the density profiles of haloes harbour no information on
their past aggregation history. This would explain why major mergers do not alter
the typical density profile of virialised objects formed by SI and do not invalidate the
peak formalism based on such a formation.
Key words: methods: analytic — galaxies: haloes — cosmology: theory — dark
matter — haloes: density profiles
1
INTRODUCTION
In the last two decades, observations and, particularly simulations, have provided us with detailed information on the
structure and kinematics of bottom-up hierarchically assembled virialised dark matter haloes. However, from the theoretical viewpoint, the situation is far from satisfactory. The
way all these properties settle down remains to be elucidated
and their connection with the power-spectrum of density
perturbations is unknown.
The accurate modelling of virialised self-gravitating collisionless dissipationless systems is an old unresolved issue. Most efforts have focused on determining the equilibrium density profile for objects formed from the monolithic collapse of an isolated spherically symmetric seed
with outward-decreasing density profile and pure Hubble
velocity field, the so-called Secondary Infall (SI) model.
Following the seminal work by Gunn & Gott (1972), the
power-law density profile was derived under the selfsimilar approximation and/or making use of an adiabatic
⋆
E-mail: [email protected]
c 0000 RAS
invariant during virialisation, both for pure radial orbits (Gunn 1977; Fillmore & Goldreich 1984; Bertschinger
1985; Hoffman & Shaham 1985) and non-radial orbits
(Ryden & Gunn 1987; White & Zaritsky 1992; Nusser 2001;
Le Delliou & Henriksen 2003). The departures from spherical symmetry (by adopting, instead, cylindrical symmetry; Ryden 1993; Moutarde et al. 1995) and self-similarity
(Lokas 2000; Lokas & Hoffman 2000) were also investigated.
These theoretical results were tested and complemented by the information drawn from specifically designed numerical tools Gott (1975); Williams et al. (2004)
as well as full cosmological N -body simulations (e.g.
Frenk et al. 1985; Quinn et al. 1986; Efstathiou et al. 1988;
Zaroubi et al. 1996). One important finding along this latter line was that cold dark matter (CDM) haloes with different masses show similar scaled spherically averaged density profiles (Dubinski & Carlberg 1991; Crone et al. 1994).
Navarro et al. (1997) showed that they are well-fitted down
to about one hundredth of the total radius by a simple analytic expression, the so-called NFW profile, that deviates
from a power-law. This finding opened a lively debate about
the value of the central asymptotic behaviour of the halo
2
Salvador-Solé at al.
density profile (e.g. Fukushige & Makino 1997; Moore et al.
1998; Ghigna et al. 2000; Jing & Suto 2000; Power et al.
2003; Hayashi et al. 2004). More recently, the Einasto (1965)
profile was shown to give even better fits down to smaller
radii (Navarro et al. 2004, 2010; Merritt et al. 2005, 2006).
The origin of this density profile remains to be understood. Certainly, it can only arise from the way haloes
aggregate their mass, which, in hierarchical cosmologies,
is through continuous mergers. But the dynamical effect of mergers depends on the relative mass of the captured and capturing haloes, ∆M/M . For this reason, it
is usually distinguished between minor and major mergers
(∆M/M greater or less than about 0.3; Salvador-Solé et al.
2007). Major mergers have a dramatic effect each on
the structure of the object. While minor mergers contribute jointly, together with the capture of diffuse matter (if any), to the so-called accretion that yields only a
small secular effect on the accreting object. Some authors
(Syer & White 1998; Raig et al. 1998; Subramanian et al.
2000; Dekel et al. 2003) studied the possibility that the nonpower-law density profile for simulated haloes is the result
of repeated major mergers. Others (Avila-Reese et al. 1998;
Huss et al. 1999; Del Popolo et al. 2000; Ascasibar et al.
2004; MacMillan et al. 2006) concentrated on the effects of
pure accretion (PA), that proceeds according to the simple SI model above from peaks (secondary maxima) in the
primordial random Gaussian density field (Doroshkevich
1970; Bardeen et al. 1986). The density profile found in this
latter scenario appears to be similar, indeed, to that of
simulated haloes. Furthermore, recent cosmological simulations have confirmed that major mergers play no central
role in setting the structure and kinematics of dark matter
haloes (Wang & White 2009). However, the reason why major mergers would not alter the density profile set by PA is
not understood.
In the present paper, we develop an accurate model of
halo density profile that clarifies all these issues. For simplicity, we consider pure dark matter systems, that is we
neglect the effects of baryons on halo structure. This will allow us to directly compare the theoretical predictions of the
model to the results of N -body simulations. This model is
built within the SI framework and uses the peak formalism,
which assumes there is a one-to-one mapping between haloes
and the density peaks. However, peaks are not assumed to be
spherically symmetric and endowed with their typical filtered
density profile calculated by Bardeen et al. (1986, hereafter
the BBKS profile) as usual, but we take into account that
they are triaxial (Doroshkevich 1970; Bardeen et al. 1986)
and consider their accurate spherically averaged unconvolved
density profile. In addition, we account for the initial peculiar velocities believed to affect the central halo density
profile (Hiotelis 2002; Ascasibar et al. 2004; Mikheeva et al.
2007). Finally, instead of making use of the usual adiabatic
invariant, poorly justified near turnaround and greatly complicating the analytic derivation of the final density profile,
we take advantage of the fact that accreting haloes develop
outwardly, keeping their instantaneous inner structure unaltered. The validity of such an assumption, hereafter simply referred to as inside-out growth, is accurately checked
by verifying that the trends observed in simulated haloes
evolving by smooth accretion are reproduced by an analytic
model where the haloes are forced to grow inside-out.
And what about the fact that major mergers are ignored in SI? This important caveat affects not only the
modelling through SI of the halo density profile, as mentioned, but also of halo statistics in the so-called peak
formalism (Peacock & Heavens 1990; Bond & Myers 1991;
Manrique & Salvador-Solé 1995, 1996, hereafter MSSa and
MSSb, respectively; Manrique et al. 1998) which relies on
the Ansatz that there is a one-to-one correspondence between haloes and peaks as in spherical collapse1 . Although
no such one-to-one correspondence is actually found in simulations (Katz et al. 1993), this is because peaks show nested
configurations (MSSa) which are not corrected for. If one
concentrates in simply checking whether or not halo seeds,
hereafter also called protohaloes, coincide with peaks, then
the answer is positive in all but a few percent of cases
compatible with the frequency of non-fully virialised haloes
(Porciani et al 2002). However, there is still the problem
that, in principle, major mergers should blur the correspondence between peaks and haloes arising from SI. In the
present paper, we argue that the reason why halo statistics and the spherically average density profile of virialised
haloes seem to be unaffected by major mergers is that virialisation is a real relaxation process. We stress that the model
deals with fully virialised haloes. This is important because,
after a major merger, the density profile of a halo spends
some time to adopt a stable density profile.
The outline of the paper is as follows. In Section 2, we
present some general relations holding for all systems regardless of their symmetry. Taking into account these relations,
we develop in Section 3 the model for triaxial haloes grown
by PA. In Section 4, we calculate the properties of typical
protohaloes and use them in Section 5 to derive the typical
spherically averaged density profile for CDM haloes. In Section 6, we discuss the effects of major mergers. The main
results are summarised in Section 7.
Throughout the paper, we adopt the ΛCDM Wmap7
(Komatsu et al. 2011) concordance model. In spite of
this, we use Newtonian dynamics with null cosmological constant as its effects are irrelevant at the scale of
virialised haloes. A package with the numerical codes
used in the present paper is publicly available from
www.am.ub.es/∼cosmo/haloes&peaks.tgz.
2
GENERAL RELATIONS FOR SPHERICAL
AVERAGED PROFILES
In the present section, we derive some general relations for
spherically averaged quantities that hold for any arbitrary
system regardless of its symmetry and that will later be used
to build the model.
Consider a self-gravitating system with arbitrary mass
distribution, aggregation history and dynamical state. The
local density and gravitational potential can be split in the
respective spherical averages around any given point and the
corresponding residuals,
ρ(r, θ, ϕ) = hρi(r) + δρ(r, θ, ϕ)
(1)
1
The excursion set formalism is also based on a one-to-one correspondence between haloes and overdense regions in the initial
density field as in spherical collapse.
c 0000 RAS, MNRAS 000, 000–000
3
Halo Density Profile
Φ(r, θ, ϕ) = hΦi(r) + δΦ(r, θ, ϕ) .
(2)
The spherically averaged gravitational potential,
hΦi(r) =
Z
1
4π
2π
Z
π
dϕ
0
dθ sin θ Φ(r, θ, ϕ),
(3)
0
satisfies, by the Gauss theorem, the usual Poisson integral
relation for spherically symmetric systems,
GM (r)
dhΦi(r)
=
,
dr
r2
where M (r) is the mass within r,
Z
M (R) = 4π
(4)
R
dr r 2 hρi(r) .
0
(5)
Taking into account the null spherical averages of δρ
and δΦ (see eqs. [1]–[2]), the potential energy within the
sphere of radius R,
W (R) =
1
2
Z
Z
R
dr r 2
0
2π
Z
π
dϕ
0
dθ sin θ ρ(r, θ, ϕ) Φ(r, θ, ϕ) , (6)
0
can be rewritten as
W (R) = 2π
Z
Z
= −4π
R
2
dr r [hρi(r) hΦi(r) + hδρ δΦi]
0
R
dr r 2 hρi(r)
0
Z
GM (r)
+ 2π
r
dr r 2 hδρ δΦi
≡ W(R) + δW(R) ,
(7)
where we have introduced the “spherical” potential energy
within R,
W(R) = −4π
Z
R
0
dr r 2 hρi(r)
GM (r)
.
r
(8)
The second equality in equation (7) follows from partial integration of the first term on the right of the first equality,
then application of the relation (4), and one new partial
integration, choosing the origin of the spherically averaged
potential so as to have
GM (R)
hΦi(R) = −
.
(9)
R
Note that, for spherically symmetric systems, this boundary condition coincides with considering the usual potential
origin at infinity and the system truncated at R.
The kinetic energy within R is
K(R) = 2π
Z
R
2
0
2
dr r hρi(r) σ (r) ,
(10)
being σ (r) the velocity variance in the infinitesimal shell at
r.2 Thus, the total energy of the sphere, E(R) = K(R) +
W (R), takes the form
E(R) = 4π
R
0
+ 2π
Z
0
σ 2 (r)
GM (r)
dr r 2 hρi(r)
−
2
r
(11)
where we have introduced the “spherical” total energy
Such a velocity variance coincides with the spherical average of
the local value. Therefore, equation (10) could also be written in
terms of the local velocity variance in angular brackets.
c 0000 RAS, MNRAS 000, 000–000
0
.
(12)
σ 2 (r) = s2 (r) + δs2 (r) ,
(13)
through the residual δs2 (r) to be specified, contributing to
the residual δE (r). In other words, there is some freedom in
the definition of s2 (r). We will come back to this point later.
So far we have considered a system with arbitrary mass
distribution (symmetry), aggregation history and dynamical
state. If the system is in addition in equilibrium, then multiplying the steady collisionless Boltzmann equation (e.g.
eq. [4p-2] in Binney & Tremaine 1987) by the radial particle velocity and integrating over velocity and solid angle, we
are led to
d(hρiσr2 )
hρi(r)
+
[3σr2 (r) − σ 2 (r)]
dr
r
Z 2π Z π
1
=−
dϕ dθ sin θ ρ(r, θ, ϕ) ∂r Φ(r, θ, ϕ) ,
4π 0
0
(14)
hρi(r)
d(hρiσr2 )
+
[3σr2 (r) − σ 2 (r)]
dr
r
GM (r)
= −hρi(r)
− hδρ ∂r (δΦ)i(r) ,
(15)
r2
identical to the Jeans equation for spherically symmetric
systems in equilibrium but for the last term on the right.
Multiplying equation (15) by 4πr 3 and integrating over the
sphere of radius R, the same steps leading to the scalar virial
relation for spherically symmetric self-gravitating systems
now lead to
4πR3 hρi(R)σr2 (R) − 2K
Z
= −4π
0
R
dr r 2 hρi(r)
Z
GM (r)
−4π
r
0
R
dr r 3 hδρ ∂r δΦi(r) . (16)
Therefore, defining the so-called “spherical” radial velocity
variance s2r (r) through
σr2 (r) = s2r (r) + δs2r (r) ,
with
δs2r (r)
1
= 3
r hρi(r)
Z
0
(17)
r
dr̃ r̃ 2 δs2 (r̃) − r̃hδρ ∂r δΦi(r̃) ,
(18)
the virial relation (16) takes the usual form for spherically
symmetric systems,
Z
0
R
dr r 2 hρi(r) s2 (r)−
GM (r)
, (19)
r
from now on called the “spherical” virial relation. Furthermore, defining the “spherical” scaled surface pressure term
(from now on simply spherical surface term), S(R), equal to
3
2
GM (r)
s2 (r)
−
dr r hρi(r)
2
r
4πR3 hρi(R) s2r (R) = 4π
R
dr r 2 hδρ δΦi(r) ≡ E (R) + δE (R) ,
2
Note that E (R) is written in terms of the “spherical” velocity variance s2 (r), different, in general, from the ordinary
velocity variance,
2
Z
R
where ∂r stands for radial partial derivative3 . Taking into
account equation (4), equation (14) adopts the form
R
0
E (R) = 4π
Z
To derive equation (14) it is only needed such conventional
conditions as the continuity of the local density and mean velocity
and the fact that the velocity distribution function vanishes for
large velocities.
4
Salvador-Solé at al.
the member on the left of equation (19) over the absolute
value of W(R), the spherical virial relation (19) adopts the
usual compact form
2E (R)
= 1 − S(R) .
W(R)
(20)
As can be seen, all the ordinary quantities X can be expressed in the form X = X + δX with the quantities X having the same form for spherically symmetric systems, hence
why they are labelled “spherical”. The residuals δX always
measure the deviation from sphericity of the quantities X
in the sense that, when δρ and δΦ vanish, δX also vanish
and X become equal to the spherical counterparts X and,
hence, recover the form for spherically symmetric systems.
The dimensionless quantities |δX /X | are, however, not necessarily less than one, so we should not look at the spherical
quantities X as 0th-order approximations of X in expansion
series on small deviations from sphericity; they are fully exact quantities and so are also the relations between them. In
particular, equations (5), (12) and (20) respectively giving
the mass, spherical total energy and spherical virial relation
or, equivalently, the relations
1 dM
,
4πr 2 dr
hρi(r) =
s2 (r) = 2
(21)
GM (r)
dE /dr
+
,
dM/dr
r
(22)
and
s2r (r) =
2E (r) − W(r)
,
r dM/dr
(23)
following from their differentiation are exact and hold unchanged regardless of the particular shape of the object.
Thus, the profiles hρi(r), s2 (r) and s2r (r) do not depend on
shape and can be determined by solving equations (21)–(22).
In contrast, the ordinary kinematic profiles σ 2 (r) and σr2 (r)
cannot be obtained from a similar set of equations because
they depend explicitly on the shape of the object.
We want to remark that, even if the spherical quantities
at any given moment behave as their ordinary counterparts
in spherically symmetric systems, the relation between their
values in different epochs may not be the same as in such systems. For instance, the spherical total energy inside spheres
of fixed mass may not be conserved in the absence of shellcrossing4 . Only one specific choice of s2 (r) will ensure such
a conservation. Note also that, for any arbitrary mass distribution, it might not be possible to have one only profile
s2 (r) satisfying that condition for the whole series of embedded spheres the system decomposes into. But, in the case of
a centred triaxial system, this is always possible thanks to
the ordered radial mapping of the mass distribution. From
now on, we assume such a symmetry and the conservation
of the spherical total energy in the absence of shell-crossing.
This is crucial for the model as it will allow us to fix the
density profile for a virialised object from the properties of
its seed (see Sec. 3).
4
If one chooses s2 (r) ≡ σ2 (r), the spherical total energy is not
conserved in ellipsoidal systems because any given sphere exchanges energy with the rest of the system.
3
THE MODEL
We will concentrate, in a first step, on triaxial systems evolving by PA. This includes the case of accretion along filaments.
3.1
Inside-out Growth
The model relies on the assumption that accreting objects
grow inside-out. This is consistent with the properties of simulated haloes (Salvador-Solé et al. 1998; Salvador-Solé et al.
2005; Helmi et al. 2003; Romano-Dı́az et al. 2006;
Wang et al. 2011). In Manrique et al. (2003) and
Salvador-Solé et al. (2007), it was shown that the assumption that (spherical) haloes grow inside-out at the
typical cosmological accretion rate automatically leads to a
density profile à la NFW. Nonetheless, to be fully confident
about it, we show below that such a growth is naturally
expected and in agreement with the detailed results of
simulations.
In SI, the isodensity contours in the triaxial seed
expand, without crossing each other, until they reach
turnaround. This expansion is achieved in linear regime with
all particles moving radially, so the axial ratios of shells
are conserved. However, after reaching turnaround, particles collapse and rebound non-radially, so the shape of the
isodensity contours changes. Shell-crossing causes a secular
energy transfer between shells so that particles initially lying in a turnaround ellipsoid reach their new apocentre at
a smaller radius and not simultaneously. However, the energy lost by particles in one orbit is small, so the typical
time of apocentre variation is substantially greater than the
typical orbital period, implying that particles belonging to
one turnaround ellipsoid will still trace an effective apocentre surface, which for symmetry reasons will also be an
ellipsoid although smaller in size and with different axial ratios. Repeating the same reasoning from each new apocentre ellipsoid, we have that any turnaround ellipsoid evolves
through a continuous series of ellipsoidal apocentre surfaces
that progressively shrink and change their shape until they
stabilise.
During such an evolution, apocentre ellipsoids never
cross each other. If two of them had a common point, particles belonging to the two surfaces at that point (with null
velocities) would follow the same evolving orbits, so the two
surfaces would always be in contact, but this is meaningless
because different turnaround ellipsoids come from different
isodensity contours in the seed which do not intersect. Thus,
during virialisation, the system contracts orderly, without
apocentre-crossing and when a new apocentre ellipsoid stops
shrinking, it necessarily places itself beyond all previously
stabilised apocentre ellipsoids. Thus, the central steady object necessarily grows from the inside out according to the
gradual deposition of particles with increasingly larger apocentre surfaces.
In the above reasoning, we assumed that orbits eventually stabilise. However, new shells are constantly arriving and crossing the growing halo, consequently, orbits only
approximately stabilise. It is therefore worthwhile verifying
the validity of this approximation. This can be accurately
checked by means of the detailed follow-up of the density
profile for CDM haloes in simulations. Zhao et al. (2009)
c 0000 RAS, MNRAS 000, 000–000
Halo Density Profile
5
by Zhao et al.: accreting halos moved in the Ms –rs log-log
plane along straight lines with exactly the same universal
slope (Ms ∝ rs1.65 ) as in our experiment and ended up at
z = 0 lying along another straight line, with a different
slope (Ms ∝ rs2.48 ) also identical to that found in our experiment (see their Fig. 22). The fact that our constrained
model reproduces the observed trend convincingly demonstrates haloes primarily grow inside-out.
3.2
Radius Encompassing a Given Mass
As we will show, by assuming that haloes grow inside-out
via PA and that the spherical total energy in spheres of fixed
mass is conserved in the absence of shell-crossing, we are able
to infer the radius enclosing a given mass in a halo from the
spherical energy distribution of its progenitor protohalo.
To do this, we will consider the virial relation (20)
for a perfectly uniform sphere with mass M , which implies
W(M ) = −3GM 2 /(5R), with spherical total energy E (M )
equal to that of the system at turnaround Eta (M ) and null
spherical surface term S(M ),
R(M ) = −
Figure 1. Tracks followed in the Ms –rs plane by typical accreting
haloes developing inside-out, with current masses, M , equal to
1011 , 1012 , 1013 , 1014 and 1015 M⊙ (from bottom to top) at z = 0,
2, 4, 6 and 8 (circles essentially from right to left, respectively in
black, green, red, blue and magenta). The straight dashed red line
shows the universal direction followed by such tracks, identical to
that found by Zhao et al. (2009) in full cosmological simulations.
The black straight line shows the Ms –rs relation obtained at z = 0
also identical to that found in Zhao et al. al. (2009; see Fig. 22).
and Muñoz-Cuartas et al. (2011) carried out such a followup and found that the rs and Ms NFW shape parameters5
of accreting haloes are not constant but vary slightly with
time. This was interpreted as evidence that the inner structure of accreting haloes is changing. But this is not the only
possible interpretation. As the density profile of haloes is
not perfectly fitted by the NFW profile6 , even if the inner
density profile did not change, its fit by the NFW law out to
progressively larger radii should result in slightly different
best values of rs and Ms .
To confirm that our interpretation for this change is
correct we have followed in the Ms –rs log-log plane the evolution of accreting halos forced to grow inside-out at the
typical accretion rate given by the excursion set formalism according to Salvador-Solé et al. (2007) model. The result, in the same cosmology as used in Zhao et al. (2009),
is shown in Figure 1. Even if, by construction, haloes grow
inside-out without changing their instantaneous inner structure, when their density profile is fitted by a NFW profile,
they are found to move in the Ms –rs log-log plane along
one fixed direction over a distance that increases with current halo mass. This behaviour is identical to that found
5
Ms is defined as the mass within the scale radius rs where the
logarithmic slope of the density profile is equal to −2.
6 The larger the mass, the closer the density profile is to a powerlaw with index ∼ −1.5-2 (Tasitsiomi et al. 2004).
c 0000 RAS, MNRAS 000, 000–000
3 GM 2
.
10 Eta (M )
(24)
This equation is the often used estimate for the radius encompassing mass M in spherical systems with an unknown
internal mass distribution (Bryan and Norman 1998). In
principle, R(M ) is not believed to give an exact measure
of that mass in the real virialised object as this has nonuniform density profile, its spherical total energy is equal to
E (M ) instead of Eta (M ) and its surface term is not null. Yet,
as shown below, as a consequence of the inside-out growth
of accreting objects formed by PA, the inaccuracies above
exactly cancel and both radii turn out to fully coincide.
To see this, we will deform the system since its shells
reach turnaround so as to construct a virialised toy object satisfying the conditions leading to equation (24).
This is only possible in PA where the equivalent radius
of turnaround ellipsoids increases with increasing time and
the central virialised object grows inside-out. Shells reaching turnaround can then be virtually contracted one after
the other without any crossing so as to match the mass profile M (r) (although not necessarily the ellipsoidal isodensity
contours) of the real virialised object developed until some
time t. By “virtual” motion we mean a motion of shells that
preserves their particle energy and angular momentum, but
is disconnected from the real timing of the system. Of course,
such a motion will not recover at the same time the axial ratios of the isodensity contours of the virialised object, but we
only need to recover the mass profile, which by conveniently
contracting each new shell is guaranteed (there is one degree
of freedom: the final equivalent radius of the contracted ellipsoidal shell and one quantity to match: the mass within
the new infinitesimally larger radius). By construction, the
new toy object so built has the same mass profile M (r) (although not the total energy profile E(r) due to the different
ellipsoidal mass distribution) as the virialised object. But
the spherical total energy in such a toy object is equal to
that at turn-around, Eta [M (r)], as there has been no shellcrossing and E is conserved for the appropriate choice of
s2 (r) (see Sec. 2).
Of course, such a toy object is not in equilibrium. But
6
Salvador-Solé at al.
e
e
the quantity Eta (R) − W(R),
with W(R)
equal to the potential energy of a homogeneous sphere with mass M , is
positive (see eq. [25] below). Therefore, we can virtually expand each ellipsoidal isodensity contour of the toy object,
from the centre out to the edge avoiding shell-crossing, so
as to end up with a uniform density equal to the mean density ρ̄(R) of the real object inside R and still have an excess
of spherical kinetic energy. Thus, this can be re-distributed
over the sphere and the radial and tangential components
of the spherical velocity variance, e
s(r), can be locally exchanged so as to satisfy the spherical virial relation (19) at
every radius. This is again possible to achieve because there
are two degrees of freedom, e
s(r) and e
sr (r), and two conditions to fulfil, the spherical total kinetic energy in excess
and the spherical virial equation. In this way, we have built
a steady homogeneous toy object with radius, mass and total energy respectively equal to R, M and Eta (M ) and with
null value of e
sr (R) (there is no density outside the sphere
because shells having not reached turnaround at t have not
been contracted). Hence, this uniform toy object satisfies
the spherical virial relation
2Eta (R)
e
W(R)
= 1,
(25)
e
which, given the equality W(R)
= −3GM 2 /(5R), implies
equation (24).
The exact equation (24) allows one to determine the
spherically averaged density profile of the virialised object
from the properties of its seed. To do this we must simply
take into account that, during the initial expanding phase
of shells, there is no shell-crossing, so the spherical total energy in spheres with fixed mass is conserved. We can then
replace, in equation (24), Eta (M ) by its value in the protohalo, Ep (M ), and obtain, by inversion of r(M ), the mass
profile M (r) and, through equation (21), the spherically averaged density profile hρi(r) of the virialised object from the
spherical total energy distribution in the seed.
Furthermore, as the functions hρi(r), s2 (r) and s2r (r) do
not depend on the shape of the object, we can think in the
spherically symmetric case to infer the latter two functions
from the former one. In such objects, orbits are purely radial because they collapse and virialise radially. We therefore
have s2 (r) = s2r (r). Equations (22) and (23) then lead to a
differential equation for E (r) that can be readily integrated
for the boundary condition E = 0 at r = 0, the result being
E (R) = −R
Z
0
R
W(r)
dr 4π hρi(r) GM (r) +
2r 2
.
(26)
Once hρi(r) is known, we can calculate W(r) (eq. [6] and
then E (R) (eq. [26]), and apply any of the two relations (22)
and (23) to infer the profile s2 (r). But the interest of this
expression is only formal because, as mentioned, the profile
s2 (r) is not observable, not even in the spherically symmetric case where s(r) = σ(r) and sr (r) = σr (r). The reason
for this is that objects formed by PA are never spherically
symmetric. The tidal field of surrounding density fluctuations causes spherically symmetric seeds to undergo ellipsoidal collapse (Zeldovich 1970). Even if such a tidal field
is artificially removed, the system suffers radial orbit instability, also leading to triaxial virialised objects (Huss et al.
1999; MacMillan et al. 2006) .
The expression (26) is nonetheless useful to compute
the spherical total energy dissipation factor since the time
of the protohalo, D(M ) ≡ E (M )/Ep (M ), with Ep (M ) given
by equation (24). This has the following interesting implication. In objects formed by PA, there is always some spherical energy loss through shell-crossing during virialisation,
so D(M ) is greater than one. Taking into account equations
(20) and (25), this dissipation factor can be written in the
form
D(M ) =
W(M )
e )
W(M
[1 − S(M )] .
(27)
Thus, if D(M ) is greater than one, the ratio
W(M )
5
=
1+
6
e
W(M )
Z
R(M )
0
r 4 ρ̄2 (r)
dr
R(M ) R4 ρ̄2 (R)
,
(28)
must also be greater than one7 . By differentiation and after
e ) is
a little algebra it can be seen that ratio W(M )/W(M
greater than one provided only ρ̄(r) is outward-decreasing.
Consequently, we are led to the conclusion that the spherically averaged mean inner density profile of any collisionless dissipationless virialised object formed hierarchically
(through PA but also through major mergers; see Sec. 6)
is necessarily outward-decreasing.
4
PROTOHALOES
According to the previous results, to calculate the spherically averaged density profile for a CDM halo grown by PA
we only need the spherical energy distribution, Eta (M ), of
its seed, namely a peak in the initial density field filtered at
the mass scale of the halo. The cosmic time ti where the initial density field has to be considered is arbitrary although
small enough for the protohalo to be in linear regime.
Such a spherical energy distribution is given, in the
parametric form, by the spherical total energy of the protohalo,
Ep (Rp ) = 4π
×
Z
Rp
0
drp rp2 hρp i(rp )
σp2 (Rp )
[Hi rp − vp (rp )]2
GM (rp )
+
−
2
2
rp
,
(29)
in centred spheres of radii Rp encompassing the masses M ,
M (Rp ) = 4π
Z
0
Rp
drp rp2 hρp i(rp ) .
(30)
In equations (29) and (30), Hi is the Hubble constant at the
cosmic time ti of the protohalo, hρp i(rp ) is its spherically
averaged density profile, vp (rp ) is the peculiar velocity, to
0th-order in the deviations from sphericity, due to the gravitational pull by the mass excess within the radius rp and
σp (Rp ) is the uniform and isotropic background peculiar velocity dispersion inside the sphere of radius Rp 8 .
The peculiar velocity vp (rp ) is (e.g. Peebles 1980),
e ) is positive (see eq. [28]), the factor 1 − S(M)
As W(M )/W(M
must also be positive and less than one.
8 The starting value of the spherical total energy can be computed taking s2p ≡ σp2 .
7
c 0000 RAS, MNRAS 000, 000–000
7
Halo Density Profile
2G M (rp ) − 4πrp3 ρi /3
vp (rp ) =
,
3Hi rp2
(31)
being ρi the mean cosmic density at ti 9 . Bringing this expression of vp (rp ) into equation (29) and neglecting the second
order term in the perturbation with vp2 (rp ) (the term with
σ 2 (Rp ) may be greater than this owing to the contribution
2
of σDM
(ti )), we are led to
Ep (Rp ) =
σp2 (Rp )
5 H
Ep (Rp ) + M (Rp )
,
3
2
(32)
where EpH (Rp ) stands for the spherical total energy of the
protohalo in the case of pure Hubble flow (i.e. eq. [29] with
null vp and σp ). In principle, σp2 (Rp ) has two contributions:
one arising from the dark matter particle velocities at decou2
pling, adiabatically evolved until the time ti , σDM
(ti ), and
another one caused by random density fluctuations. Following the exact prescription by Mikheeva et al. (2007) for this
latter contribution, we arrive at
Z
∞
h
i
2 2 2
dk
P (k) 1 − e−k Rp
2
2π
0
2
√
2
2
2
= σDM
(ti ) +Σ2i σ−1
(0) − 2σ−1
(Rp ) + σ−1
( 2Rp ) , (33)
√
where Σi is defined as 2πHa2i , being ai the initial cosmic
2
scale factor10 , and σ−1
(Rf ) is the spectral moment of order
−1 for the power-spectrum, P (k), with the j-order moment
given by
σp2 (Rp )
σj2 (Rf )
=
=
2
σDM
(ti )
Z
0
∞
+
Σ2i
2
2
dk
P (k) k2j+2 e−k Rf .
2π 2
(34)
But the velocity variance due to random density fluc2
tuations, σp2 (Rp ) − σDM
(ti ) (see eq. [33]) turns out to be
several orders of magnitude less than GM (Rp )/Rp (a factor
∼ 10−6 in the concordance model adopted here). Consequently, it can be neglected in Ep (M ) (eq. [29]). Contrary to
the claims made by Hiotelis (2002), Ascasibar et al. (2004)
and Mikheeva et al. (2007) that such a velocity dispersion
may affect the central density profile for CDM haloes. The
former two authors reached this conclusion by analysing the
effects of velocity dispersions that, unlike those used here,
were not consistently inferred from the random density fluctuations. Mikheeva et al. used the same derivations for the
dispersion as used here, however, they considered the fluctuations within the whole protohalo, whereas we account
for random density fluctuations within each sphere of radius Rp . In including fluctuations from the whole protohalo, Mikheeva et al. included velocities induced by external density fluctuations with masses possibly larger than
the own mass of the sphere. This overestimates the velocity dispersion in the sphere because external density fluctuations cause it a bulk velocity, not a velocity dispersion.
And the bulk velocity of a sphere does not hamper its collapse, whereas the internal velocity dispersion does, as found
by Mikheeva et al. for small enough Rp . The only velocity
dispersion that does not diminish with decreasing Rp and,
hence, can really set a minimum halo mass is that intrinsic
9
In equation (31), we have taken into account that the cosmic
virial factor f (Ω) ≈ Ω0.1 is at ti very approximately equal to one.
10 At small t , this is very approximately equal to the cosmic
i
growth factor.
c 0000 RAS, MNRAS 000, 000–000
2
of dark matter particles, σDM
(ti ). But this is negligible in
CDM cosmologies.
And what about the spherically averaged peak density profile, hρp i(rp )? The BBKS profile calculated by
Bardeen et al. (1986) gives the typical (average) spherically averaged density contrast profile for peaks filtered with
a Gaussian window. Unfortunately, the convolution by a
Gaussian window results in some information loss so the
BBKS profile cannot be used to infer the desired profile. In
addition, the average profile of purely accreting haloes with
M at t may not coincide with the profile arising from the
average profile of the corresponding seeds. However, we can
still find the typical (in the sense below) spherically averaged
density profile of protohaloes.
As in PA haloes grow inside-out, the density profile inside every inner ellipsoid exactly matches that of one halo
ancestor, also evolved by PA from a peak with its own density contrast and scale. Hence, the typical (average) spherically averaged density profile for purely accreting haloes
with M at t must arise from a protohalo whose spherically
averaged density contrast profile, hδp i(rp ), convolved with
a Gaussian window of radius Rf corresponding to the mass
of any halo ancestor yields, at rp = 0, the density contrast
δpk (Rf ) of the peak evolving into that ancestor,
δpk (Rf ) =
21/2
1/2
π Rf −3
Z
∞
drp rp2 hδp i(rp ) exp −
0
1
2
rp
Rf
2 . (35)
According to the peak formalism (see Sec. 6), the typical
(most probable) trajectory δpk (Rf ) of peaks evolving into
the series of halo ancestors ending in a halo with M at t is
the solution of the differential equation (SSb)
dδpk
= −xe (δpk , Rf ) σ2 (Rf ) Rf ,
dRf
(36)
with the boundary condition defined by the halo, that is
satisfying the relations
δpk (t) = δc (t)
G(ti )
G(t)
(37)
and
Rf (M ) =
1
q
3M
4πρi
1/3
.
(38)
where G(t) is the cosmic growth factor, q = 2.75 is the radius, in units of Rf , of the collapsing cloud with volume equal
to M/ρi , δc [t(z)] = 1.93 + (5.92 − 0.472z + 0.0546z 2 )/(1 +
0.000568z 3 ) is the critical linearly extrapolated density contrast for halo formation at the redshift z (see Sec. 4 for
those values of q and δc [t(z)]). In equation (36), xe (δpk , Rf )
is the inverse of the average inverse curvature x (equal to minus the Laplacian over σ2 ) for the distribution of curvatures
(BBKS),
D E
(2π)−1/2
1
(Rf , δpk ) =
x
(1 − γ 2 )1/2
Z
∞
dx
0
2
(x−x⋆ )
−
1
f (x) e 2(1−γ 2 ) ,
x
(39)
at peaks with δpk and Rf , being
x3 − 3x
f (x) =
2
2
+
5π
1/2
erf
1/2 5
2
x + erf
8 − 5x82
31x2
+
+
e
4
5
1/2 5
2
x
2
x2
8 − 5x22
−
e
, (40)
2
5
8
Salvador-Solé at al.
Figure 2. Typical δpk –Rf (in physical units) peak trajectory
of peaks at z = 100 (solid lines) giving rise to the series of halo
ancestors evolving by PA into haloes with current masses M equal
to 10, 1 and 0.1 times the critical mass for collapse, M∗ = 3.6 ×
1012 M⊙ (curves from top to bottom, respectively in orange, red
and brown) .
and σ2 (Rf ) the second order spectral moment, where γ and
x⋆ are respectively defined, in terms of the spectral moments, as σ12 /(σ0 σ2 ) and γδpk /σ0 . This distribution function is a very peaked, quite symmetric, bell-shaped function
so that the function Rf (δpk ) inverse of the δpk (Rf ) solution
of equation (36) is traced by peaks with the average (essentially equal to the most probable) inverse curvature at
each point (δpk , Rf ). Note that the slope dRf /dδpk translates into the typical (average or most probable) accretion
rate, dM/dt, of haloes with M (Rf ) at t(δpk ) evolving from
those peaks. In Figure 2, we show the typical peak trajectories at z = 100 leading to typical haloes with current masses
equal to 10−1 M∗ , M∗ and 10M∗ , where M∗ is the critical
mass for collapse in the concordance model (3.6 × 1012 M⊙ ).
Given a typical peak trajectory, δpk (Rf ), equation (35)
is a Fredholm integral equation of first kind for hδp i(rp ).
Through the changes y = rp2 and x = 1/(2Rf 2 ), it takes the
form of a two-sided Laplace integral transform,
g(x) =
Z
∞
dy f (y) e−xy ,
(41)
−∞
with
and g(x) respectively equal to y 1/2 hδp i(y 1/2 ) and
√ f (y) −3/2
2 2π(2x)
δpk [(2x)−1/2 ], which can be solved in the standard way. Extending x to the complex space and taking
x = i2πξ, equation (41) adopts the form of a Fourier Transform,
g(ξ) =
Z
∞
dy f (y) e−i2πξy ,
(42)
−∞
√
being g(ξ) = 2 2π(i4πξ)−3/2 [δR (ξ) + iδI (ξ)], where δR (ξ)
Figure 3. Spherically averaged unconvolved density profiles for
the seeds of typical purely accreting haloes with the same current
masses M as in Figure 2 (same colours, now ordered from bottom
to top), obtained by inversion of the peak trajectories shown in
that Figure. Rp is the radius of the protoobject with mass M . The
filtering radii Rf /Rp used in the convolutions shown in Figure 4
are marked with arrows.
and δI (ξ) stand respectively for the real and imaginary parts
of δpk (ξ) ≡ δpk [(4πξ)−1/2 ]. Thus, equation (35) can be inverted by simply taking the inverse Fourier transform of
equation (42).
As f (y) is a real function, the real and imaginary parts
of g(ξ) must be even and odd, respectively. Both conditions
must be satisfied at the same time, so there are only two possibilities: either δR (ξ) = δI (ξ) or δR (ξ) = −δI (ξ). The latter
possibility leads to a nonphysical solution, so g(ξ) must be
a pure imaginary odd function, i.e. δR (ξ) = δI (ξ). To determine δpk (ξ) we must solve the differential equation (36) in
the complex space for ξ, that is for its real and imaginary
parts separately. But, as the two solutions δR and δI are
identical, we can directly solve
√ it in the real space for Rf
and then take δR = δI = δpk / 2, after the conversion from
Rf to ξ. Once the solution δpk (ξ) is known, we can readily
calculate the function g(ξ) and take its inverse Fourier transform, which leads to the wanted protohalo density contrast
profile, hδp i(rp ), after the change rp2 = y.
The spherically averaged unconvolved protohalo density
profiles, hρp i(rp ), obtained from the typical peak trajectories
depicted in Figure 2 are shown in Figure 3. In Figure 4, we
plot the convolved density contrast profiles for the seeds
of three arbitrary ancestors (with masses M (Rf ) given by
eq. [38] for the three arbitrary filtering radii) of the halo
with final mass M∗ , corresponding to the filtering radii in
units of the protohalo radius, Rf /Rp , marked with arrows in
Figure 3. For comparison, we also plot the BBKS profiles for
peaks with identical central density contrasts and filtering
radii. As can be seen, each couple of curves is very similar;
c 0000 RAS, MNRAS 000, 000–000
Halo Density Profile
Figure 4. Spherically averaged density contrast profiles for the
seed at z = 100 of a halo with current mass equal to M∗ (in red
in Figs. 2 and 3) convolved by a Gaussian window (solid lines)
with the filtering radii quoted (in physical units) and marked
with arrows in Figure 3 (same colours), compared to the BBKS
profiles with identical central density contrasts and filtering scales
(dashed coloured lines). Ranc
are the radii of the seeds of the
p
corresponding halo ancestors.
the small deviation observed (of less than 10 % within the
radius Rpanc of each ancestor seed) is likely due to round-off
errors11 . Thus, there is no evidence that the peak leading
to an object with the average spherically averaged density
profile for haloes with M at t is significantly different from
the average peak leading to such haloes.
Interestingly, the peak trajectories converge to a finite
value (null asymptotic slope) as Rf approaches to zero, as
can be seen after some algebra from the null asymptotic
limit of the right hand-side member of equation (35). This
in turn implies, from equation (35) and the relation between
hδρp i(rp ) and hρp i(rp ), that the unconvolved density profiles
for protohaloes with different masses also converge to a finite
value. This is consistent with the behaviour at small radii
of the unconvolved density profiles of protohaloes shown in
Figure 3. This finite limit is approached slightly more rapidly
than in the case of typical peak trajectories.
5
HALO DENSITY PROFILE
Thus, to derive the typical spherically averaged density profile of a halo with M at t we must follow the following steps:
9
1) solve equation (36) for the typical peak trajectory δpk (Rf )
leading by PA to the typical halo with M at t; 2) from such
a peak trajectory, invert equation (35) to find hδp i(rp ) and
from it the spherically averaged density profile of the protohalo, hρp i(rp ); 3) given the protohalo density profile, determine the spherical energy distribution Ep (M ) in the protohalo by means of equations (30) and (29); and 4) making
use of this latter function, invert equation (24) to find the
typical mass profile M (r) of the halo and, from it (eq. [21]),
the typical spherically averaged density profile hρi(r).
The density profiles so obtained (hereafter the theoretical profiles) for haloes with the same masses as in Figures 2–3 are compared in Figure 5 to the corresponding
NFW profiles with Zhao et al. (2009) mass-concentration
and a total halo radius equal to the virial radius Rvir of
Bryan & Norman (1998) (essentially equal to r90 ). As can
be seen, there is good agreement between each couple of
curves down to about one hundredth Rvir . The residuals
∆ log(hρi) (prediction minus NFW profile) start being positive at 0.01Rvir , tend to diminish at intermediate radii
(reaching slightly negative values for M∗ ), then increase
again at moderately large radii (except for 10 M∗ ) and finally become negative near the halo edge. All these trends
coincide with those shown by the same residuals (simulated
halo minus NFW profile) for individual haloes in the simulations by Navarro et al. (2004; Fig. 1, left panels) except for
the fact that the residuals of simulated haloes are notably
larger, as expected (the theoretical profiles are supposed to
correspond to average profiles). The only significant difference is near the halo edge, where the residuals of the theoretical profiles for low halo masses keep on decreasing for
a longer radial range (see also Fig. 7). This could be due
to the different halo radius used in both works: the radius
adopted in the present version of the model, arising from the
fit to the halo mass mass function predicted in the excursion
set formalism (see Sec. 6) is likely closer to Rvir ≈ r90 as in
Zhao et al. (2009) than to r200 as in Navarro et al. (2004).
But, at small r, the theoretical profiles become progressively shallower and increasingly deviate from the NFW
form (with constant inner logarithmic slope equal to −1).
This behaviour is also in agreement with the results of numerical simulations showing that the Einasto profile gives
a slightly better fit to the spherically averaged density profile for simulated haloes down to radii of about 0.001Rvir .
In Figure 5, we plot the Einasto profiles that better fit the
theoretical profiles down to 0.01Rvir . As can be seen, these
Einasto profiles are still reasonably close to the theoretical
profiles at radii four orders of magnitude less (r ∼ 10−6 Rvir ).
In Figure 6, the theoretical density profiles are also compared to the Einasto profiles with mass-dependent parameters according to Gao et al. (2008) and the same cosmology used by these authors (the mass-dependent Einasto parameters are not known for other cosmologies). Although
Gao et al. fitted the density profile of simulated haloes only
down to 0.05 the total radius12 , the agreement between each
couple of curves is similarly good as for the NFW profiles.
In this case, the agreement of the theoretical profiles with
11
The convolution is achieved by taking the product of the 3D Fourier transform of the unconvolved density contrast profile
and the Gaussian window. As the Fourier transform of the unconvolved density contrast profile has no compact support, some
aliasing is present.
c 0000 RAS, MNRAS 000, 000–000
12
Gao et al. (2008) used r200 . To transform to the same radius
Rvir as in Figure 5, their Einasto profiles have been extended to
r90 where we have computed the halo mass.
10
Salvador-Solé at al.
Figure 5. Spherically averaged density profiles predicted by the
present model (solid lines) for the same current masses M at Rvir
as in Figures 2 and 3 (same colours, ordered as in Fig. 2), compared to the density profiles of the NFW form (dashed lines) with
Zhao et al. (2009) concentrations fitting the average density profile of simulated haloes, both in the radial range covered by those
authors (left) and down to a radius four orders of magnitude less
(right). For comparison we also plot the fits to the predicted density profiles by an Einasto law (dotted lines). To avoid crowding,
the curves for 10 M∗ and 0.1 M∗ have been respectively shifted
upwards and downwards by a factor of 3. The lower panels show
the residuals of the theoretical curves from the NFW profiles (left)
and of the Einasto profiles from the theoretical curves (right) for
the three halo masses (same order from top to bottom).
the Einasto profiles at very small radii is not so good (although still much better than with the NFW profiles). But
this is likely due to the rather limited radial range covered in Gao et al. study. As the NFW profile gives, down
to ∼ 0.01Rvir , similarly good fits as the Einasto profile to
the density profile of simulated haloes, the results shown in
Figure 5 strongly suggest that the Einasto profile can do
much better at small radii if the Einasto parameters are
adjusted by covering a wider radial range.
Given a protohalo with inner asymptotic density profile
(5+α)
(3+α)
we
and Ep (rp ) ∝ rp
ρ ∝ rpα , so that M (rp ) ∝ rp
α
have, from equation (24), that hρi(r) ∝ r . Since the protohalo has a null logarithmic slope α (see Sec. 4), it follows
that the density profile for haloes must also have null inner
logarithmic slope. In other words, there is strictly no cusp
in the spherically averaged density profile for CDM haloes
according to the present model. However, the finite central
value is approached very slowly, in fact slightly more slowly
than the Einasto profile (see Fig. 5).
The main differences the density profiles predicted by
the present model and by Del Popolo et al. (2000) and
Ascasibar et al. (2004) models, both of which also use the
SI framework, are that we do not assume spherical seeds
Figure 6. Same as Figure 5 (including colours and order) but
for the comparison with the density profiles of the Einasto form
(dashed lines) with Gao et al. (2008) mass-dependent parameters fitting the average density profile of simulated haloes in the
ΛCDM cosmology adopted by those authors, both in the radial
range covered by Gao et al. although extended out to Rvir ≈ r90
(left) and down to the same small radii as in the right panel of
Figure 5 (right). The curves for 10 M∗ and 0.1 M∗ have been
shifted as in Figure 5. The lower panels show the residuals (same
lines) for the three halo masses (same order from top to bottom).
with convolved BBKS profiles and we make use of inside-out
growth instead of an adiabatic invariant to determine hρi(r).
The model presented here is essentially equivalent to the
Salvador-Solé et al. (2007) model. The key difference is that
Salvador-Solé et al. (2007) used the typical cosmological accretion rate onto haloes, whereas here we explicitly make
use of the typical density profile of halo seeds. However,
the typical halo mass accretion rate arises, as mentioned,
from the typical protohalo density profile, so this difference
is just a matter of presentation, motivated by the distinct
theoretical framework of the two models: the peak and the
excursion set formalisms. The only formal difference between
the two models, apart from the fact that Salvador-Solé et al.
also assumed spherical symmetry, is that the radius encompassing a given mass adopted by these authors was inferred
from equation (24) although not using the spherical total
energy of the real protoobject but of its top-hat approximation according to Bryan & Norman (1998) prescription. This
should introduce additional numerical differences between
the density profiles predicted by the two models. It is also
important to mention that the models by Del Popolo et al.
(2000) and Ascasibar et al. (2004) and Salvador-Solé et al.
(2007) include free parameters to be adjusted (the density
contrast and filtering radius of the peak in the two former
cases, and the value of ∆M/M setting the frontier between
c 0000 RAS, MNRAS 000, 000–000
Halo Density Profile
Figure 7. Spherically averaged density profile for a halo with
current total mass inside Rvir equal to M∗ predicted by the
present model (green line), Salvador-Solé et al. (2007) model
(orange line), the Ascasibar et al. (2004) model (red line) and
the Del Popolo et al. (2000) model (blue line), respectively ordered from bottom to top at the right end of each panel. The
Ascasibar et al.’s profile is only shown at radii larger than 0.1Rvir
as recovered by those authors.
minor and major mergers in the latter), while the present
model includes no free parameter at all.
In Figure 7, the spherically averaged halo density profile predicted by the present model is compared to those
predicted by the Del Popolo et al. (2000), Ascasibar et al.
(2004) and Salvador-Solé et al. (2007) models, all of them
in the concordance model here considered. As expected,
the theoretical profile predicted by the present model is
quite similar to that predicted by Salvador-Solé et al. model,
while these two profiles substantially deviate from those predicted by the remaining models. On the other hand, the two
former profiles are in better agreement with the NFW profile
with Zhao et al. (2009) mass-concentration relation. Moreover, the theoretical profile derived here is the one in best
agreement with such a NFW profile, which is remarkable
because it is also the only theoretical profile with no free
parameter to be adjusted.
6
IMPLICATIONS FOR MAJOR MERGERS
The present model has been built for haloes formed by PA.
Not only is this kind of halo formation crucial for the insideout growth condition, but also for the possibility to apply
the peak formalism in order to derive the peak trajectory
leading to a typical halo with a given mass at a given time.
The peak formalism itself is based on the existence of a oneto-one correspondence between haloes and peaks inspired
in the SI model that ignores major mergers. This therefore
c 0000 RAS, MNRAS 000, 000–000
11
raises two important questions. How do major mergers affect
the typical spherically averaged density profile for haloes
derived under these conditions? And how do they affect the
peak formalism?
As shown in Section 3, given a seed with known spherically averaged density profile, we can find the spherically
averaged density profile of the virialised halo evolving from
it by PA, but the converse is also true. Given a halo grown by
PA, we can calculate from its mass profile M (r) the spherical
total energy of the protohalo, Ep (M ) (eq. [24]), and then determine its spherically averaged density profile, hρp i(r), from
equations (29) and (30). Therefore, there is in PA a one-toone correspondence between the initial and final spherically
averaged density profiles. This is a well-known characteristic of spherical SI (Del Popolo et al. 2000), extended in the
present paper to non-spherical SI.
Interestingly, the reconstruction of the spherically averaged density profile for the seed of a halo having grown
by PA can also be applied to a halo having suffered major
mergers. This yields the spherically averaged density profile,
hρp i(rp ), of a putative peak that would evolve by PA into a
halo with a spherically averaged density profile identical, by
construction, to that of the original halo. Clearly, if the halo
has grown by PA, such a putative peak exists and it is an
ordinary peak. But if the halo has undergone major mergers,
does it exist? Is it an ordinary peak? In which halo does it
evolve? To answer these questions we will make use of the
rigorous treatment of the peak formalism given in MSSa and
MSSb.
As mentioned, the peak Ansatz at the base of the
peak formalism states that there is a one-to-one correspondence between haloes with M at t and peaks in the filtered density field at some small enough cosmic time ti ,
for some monotonous decreasing and increasing functions
δ(t) at Rf (M ) of the respective arguments. According to
this Ansatz, peaks associated with accreting haloes describe
continuous trajectories in the δpk –Rf diagram. Those peaks
need not necessarily be anchored to points with fixed coordinates; they can move as the filtering scale varies (as
real haloes do in the clustering process). But, thanks to the
mandatory use of the Gaussian window13 , the connection
can be made in a simple consistent way between peaks tracing one given accreting halo at contiguous scales14 .
In a major merger, the continuous trajectories of (connected) peaks tracing the merging haloes are interrupted,
while one new continuous peak trajectory appears tracing
the halo resulting from the merger, leaving a finite gap in
Rf due to the skip in halo mass between the merging and final objects. This is the only process where peak trajectories
are interrupted. Haloes that do not merge but are accreted
by more massive haloes are traced by peaks that do not
disappear but become nested into the collapsing cloud of
larger scale peaks tracing the accreting haloes. This leads to
a complex nesting of peaks with identical δpk but different
Rf . Once such a nesting is corrected, the number density of
13 The decreasing behaviour of δ with increasing R implied by
f
the growth in time of halo masses is only guaranteed for that
particular filtering window (see MSSa).
14 A peak at scale R is connected with another peak at scale
f
Rf + dRf provided only these two peaks are at a distance smaller
than Rf from each other (see MSSa).
12
Salvador-Solé at al.
peaks with δpk at scales between Rf and Rf + dRf and its
filtering evolution recovers the mass function (and growth
rates) of virialised haloes (MSSa and MSSb).
Thanks to these results it can be shown (see MSSa)
that the δpk (t) and Rf (M ) relations defining the one-toone correspondence between (non-nested) peaks and (nonnested) haloes stated in the peak Ansatz are necessarily of
the form (37)–(38). These relations can be seen as the generalisation of those of the same form found in top-hat spherical collapse, with δc (t) and q respectively equal to 1.686
and 1. This does not mean, of course, that these parameters must take the same values in the peak formalism. On
the contrary, the freedom in δc (t) and q makes it possible
to account for the change in the filtering window (Gaussian instead of top-hat) and possibly also in the departure
from spherical collapse in the real clustering process15 For
δc [t(z)] = 1.93 + (5.92 − 0.472z + 0.0546z 2 )/(1 + 0.000568z 3 )
and q = 2.75, the halo mass function predicted in the ΛCDM
concordance cosmology (after correction for nesting according to MSSa) recovers the mass function derived from the
excursion set formalism from z = 0 up to any arbitrarily
large z 16 . As the halo mass function at t predicted in the
peak formalism is nothing but the filtering radius distribution for peaks with δpk (t) (eq. [37]) transformed into the
former by means of the relation (38), this result implies that
there is indeed a one-to-one correspondence between peaks
and haloes as stated by the peak Ansatz.
The putative peak of a halo and its associated peak according to the peak Ansatz have the same δpk (t) and Rf (M ).
Their filtering evolution may be different: the trajectory in
the δpk –Rf diagram of the putative seed can always be traced
down to any arbitrarily small filtering radius, whereas the
trajectory of the associated peak can only be traced until
reaching the filtering radius corresponding to the last major
merger. However, the values δpk and Rf of a peak do not allow one to tell its ‘past’ filtering evolution because there is, in
the filtering process, a kind of ‘memory loss’. The Gaussian
window, mandatory as mentioned for the peak formalism,
yields a strong correlation between very close scales, which
is welcome to carry out the connection between peaks at
contiguous scales. But, at the same time, it yields a loss of
correlation between scales different enough to encompass the
gap produced in major mergers. Thus, owing to this particular window, peaks at a given scale do not know whether or
not they have appeared in some major merger (at a smaller
scale). Therefore, the putative seed of a halo coincides with
its associated peak according to the peak Ansatz and, as
such, it is an ordinary peak. (It contributes to the filtering
scale distribution of peaks with a given δpk , regardless of the
past history of the halo.)
15
For this latter aspect to be true, we should adjust the mass
function obtained form simulations or from the excursion set formalism with values of δc and q better adjusting the former mass
function. In the present paper we fit, however, the mass function
predicted in the excursion set formalism with δc (t) = 1.686 and
q = 1, so the departure from non-spherical collapse cannot be
accounted for.
16 The mass function predicted by the excursion set formalism is
not fully accurate, particularly at large z, so one should rather fit
the mass function drawn from numerical simulations. But this is
not important here.
Thus, according to the present model supported by the
results shown in Section 5, the spherically averaged density
profile of a halo having undergone major mergers must be
indistinguishable from that arising from the evolution by
PA of an ordinary peak (the putative peak) or, equivalently,
there must be in the halo aggregation process a memory
loss similar to that mentioned above affecting the filtering
process of peaks. Such an implication has to do with the
fundamental debate on whether or not virialisation is a real
relaxation (e.g. Henriksen 2009).
As noticed by Del Popolo et al. (2000), the one-to-one
mapping between the initial and final density profiles in PA
seems to indicate that there is no memory loss, at least in
PA, during virialisation. This is at odds, however, with the
fact that virialisation, even in PA, sets a time arrow. Even
though the equations of motion of individual particles are
time-reversible, owing to the highly non-linear dynamics of
shell-crossing, any infinitesimal inaccuracies in the positions
and velocities of particles in a simulation are rapidly amplified, causing simulated orbits to deviate from the true ones.
When the simulation is run forwards, this goes unnoticed
because the system always reaches the same (statistically
indistinguishable) final equilibrium configuration. However,
when the simulation is run backwards it is readily detected:
before shells can reach the maximum size (at turnaround),
they begin to contract again towards the final equilibrium
state. As the existence of a time arrow (or the impossibility
of reversing by numerical means the evolution of a system) is
an unambiguous signature of relaxation, we must conclude
that virialisation is a real relaxation.
In the case of PA the memory loss on the initial conditions is not complete: the order of shell apocentres is conserved. Consequently, despite the fact that information regarding individual orbits is lost, we can recover the initial
configuration in a statistical sense (in that we can recover
the density profile). Thus, even in this case, the initial configuration cannot be exactly recovered because of the loss
of information on the phase of particle orbits (individual orbits are mixed up). And, in the general case including major
mergers, the relaxation nature of virialisation is even more
evident. The initial configuration of a given virialised object can never be recovered, not even in a statistical sense,
because the density profile for virialised haloes does not harbour information on their past history, so we can never be
sure that the unconvolved density profile of their putative
seeds, derived assuming PA, describes the density profile of
their real seeds.
The prediction made by the present model that the
spherically averaged density profile of haloes does not harbour information on their past aggregation history is thus
consistent with the general behaviour of virialisation. Moreover, it is consistent with several specific results regarding
the density profile of simulated virialised haloes: i) virialised
haloes with very different aggregation histories have similar
spherically averaged density profiles (e.g. Nusser & Sheth
1999), ii) their density profile is independent of the time
they suffered their last major merger (Wechsler et al. 2002)
and iii) their density profile does not allow one to tell how
many, when and how intense the major mergers they have
suffered are (Romano-Dı́az et al. 2006). Certainly, there are
also some claims in the literature that the density profile
for haloes depends on their formation time. But this is due
c 0000 RAS, MNRAS 000, 000–000
Halo Density Profile
to the particular definition of halo formation time adopted
in those works, related to the form of the profile. Thus, the
prediction of the model regarding the lack of information
in the density profile of a halo on its past history is not
contradicted by the results of numerical simulations.
Thus, the present model strongly suggests that haloes
having suffered major mergers have density profiles which
are indistinguishable from those that have not. This prediction is important not only for the validity of the model (i.e.
the validity of the inside-out growth assumption and the
peak Ansatz), already supported by the good behaviour of
the density profiles it predicts (Sec. 5), but also because it
allows one to understand why major mergers do not affect
neither the typical halo density profile nor the peak formalism for halo statistics based on SI. Indeed, the fact that
haloes formed in a major merger have, after complete virialisation, would have a density profile indistinguishable from
that of a halo formed by PA justifies that any halo with M
at t can be associated with a peak with δpk and Rf as if it
had evolved from it by PA, as assumed in the peak Ansatz
and that the typical halo density profile derived assuming
PA is not affected by major mergers. As a corollary we have
that the present model of spherically averaged density profile should hold for all haloes, regardless of their aggregation
history.
7
SUMMARY
The present model relies on two assumptions: i) that the typical density profile for virialised objects in hierarchical cosmologies with collisionless dissipationless dark matter can
be derived as if all the objects grew by PA and ii) that the
peak formalism correctly describes halo statistics. The key
points of the derivation are that, contrarily to the kinematic
profiles, the spherically averaged density profile does not depend on the (triaxial) shape of the object and that, during
accretion, virialised objects grow from the inside out, keeping the inner structure essentially unchanged.
Triaxial collisionless dissipationless systems undergoing
PA virialise by transferring energy from inner to outer shells
through shell-crossing. Due to this energy loss, particles progressively reaching turnaround describe orbits that contract
orderly in the sense that the ellipsoidal surfaces effectively
traced by their apocentres shrink and change their axial ratios without crossing each other until they stabilise. This
causes the central steady object to develop inside-out. Such
an evolution of accreting objects is in full agreement with
the results of numerical simulations.
One important consequence of the inside-out growth is
that the radius encompassing a given mass in triaxial virialised objects formed by PA exactly coincides with the usual
estimate of such a radius from the energy of the sphere
with identical mass at turnaround. In this conditions, there
is a one-to-one mapping between the density profile of the
virialised object and its seed. This allows to infer the typical spherically averaged density profile, hρi(r), of virialised
objects from the energy distribution of their spherically
averaged seeds, which can be calculated from the powerspectrum of density perturbations making use of the peak
formalism.
The consistency between the peak formalism and the
c 0000 RAS, MNRAS 000, 000–000
13
one-to-one mapping between the initial and final density
profiles following from PA implies that virialisation must
be a real relaxation. This conclusion agrees with the general
behaviour of virialisation (it sets a time arrow) as well as
with specific results regarding the density profile of simulated haloes. As such, the spherically averaged density profile for haloes cannot harbour information on their past aggregation history which explains why major mergers do not
alter the typical density profile of virialised objects derived
under the PA assumption and do not invalidate the peak
Ansatz. In other words, the sole condition that virialisation
is a real relaxation is enough for the two assumptions of the
model to be not only consistent but also fully justified.
The model has been applied to CDM haloes. We have
derived the typical unconvolved spherically averaged density profile for peaks evolving by PA and from it the typical
spherically averaged density profile for haloes with a given
mass in a given epoch. Specifically, we have established the
link between the typical halo density profile and the powerspectrum of density perturbations in any given hierarchical
cosmology. The typical halo density profile so predicted in
the ΛCDM concordance cosmology is in very good agreement with the NFW and the Einasto profiles fitting the
spherically averaged density profile of simulated haloes down
to one hundredth the total radius. However, such a theoretical profile does not have a central cusp. In the ΛCDM
cosmology, the model predicts that simulations reaching increasingly higher resolutions will find a typical halo density
profile that tends to a core (null asymptotic slope) which is
slowly approached, even slower than current fits using the
Einasto profile.
ACKNOWLEDGEMENTS
This work was supported by the Spanish DGES, AYA200615492-C03-03 and AYA2009-12792-C03-01, and the Catalan
DIUE, 2009SGR00217. JV was beneficiary of the grant BES2007-14736 and SS of a grant from the Institut d’Estudis
Espacials de Catalunya. We acknowledge Gary Mamon
and Guillermo González-Casado for revising the original
manuscript and an anonymous referee for his constructive
criticism.
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